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	Volume 1, Issue 2 Pg 101 - 174 (June 2009)
	Volume 1, Issue 1 Pg 1 - 100 (March 2009)

International Journal of Numerical Methods and Applications

International Journal of Numerical Methods and Applications
Volume 1, Issue 2, Pages 155 - 174 (June 2009)

ON WU AND SCHABACK’S ERROR BOUND
Lin-Tian Luh (Taiwan)

Abstract:
Radial basis functions are a very powerful tool in multivariate approximation since they are mesh free. In the theory of radial basis functions, the most frequently used error bounds are the one raised by Madych and Nelson in [5] and the one raised by Wu and Schaback in [7], especially the latter. It seems that Wu and Schaback’s error bound is five times more frequently used than that of Madych and Nelson’s. The reason is that Wu and Schaback’s space of approximated functions is much easier to understand, and their error bound is much easier to use. Unfortunately, this error bound contains crucial mistakes which will be pointed out in this paper. Moreover, the r.b.f. people have misunderstood its space of approximands for long. This will also be discussed in depth. These, to some extent, may be a blow to r.b.f. people. However, Madych and Nelson’s error bound is still very powerful because it is highly related to Sobolev spaces which contain solutions of many differential equations. Moreover, Wendland’s error bound in [6] may also be used. No error has been discovered in the two error bounds hitherto. However, one should be very careful when using Wendland’s error bound because, as pointed out explicitly by Wendland in page 204 of [6], Wendland’s algebraic-type error bound is to some extent based on Wu and Schaback’s error bound. All these discussions are based on the central idea of this paper, i.e., unsmooth function should be approximated by unsmooth function.

Keywords and phrases: multivariate interpolation, Kriging function, multiquadrics, Gaussians, thin-plate splines.

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